Well-posedness and peakons for a higher-order \(\mu\)-Camassa-Holm equation. (English) Zbl 1483.35074
Summary: In this paper, we study the Cauchy problem of a higher-order \(\mu\)-Camassa-Holm equation. By employing the Green’s function of \((\mu - \partial_x^2)^{- 2}\), we obtain the explicit formula of the inverse function \((\mu - \partial_x^2)^{- 2} w\) and local well-posedness for the equation in Sobolev spaces \(H^s(\mathbb{S})\), \(s > \frac{7}{2}\). Then we prove the existence of global strong solutions and weak solutions. Moreover, we show that the data-to-solution map is Hölder continuous in \(H^s(\mathbb{S})\), \(s \geq 4\), equipped with the \(H^r(\mathbb{S})\)-topology for \(0 \leq r < s\). Finally, the equation is shown to admit single peakon solutions which have continuous second derivatives and jump discontinuities in the third derivatives.
MSC:
| 35G25 | Initial value problems for nonlinear higher-order PDEs |
| 35C08 | Soliton solutions |
| 35J08 | Green’s functions for elliptic equations |
Keywords:
higher-order \(\mu\)-Camassa-Holm equation; global existence; weak solutions; Hölder continuity; peakon solutionsReferences:
| [1] | Beals, R.; Sattinger, D. H.; Szmigielski, J., Inverse scattering solutions of the Hunter-Saxton equation, Appl. Anal., 78, 255-269, (2001) · Zbl 1020.35092 |
| [2] | Bressan, A.; Constantin, A., Global solutions of the Hunter-Saxton equation, SIAM J. Math. Anal., 37, 996-1026, (2005) · Zbl 1108.35024 |
| [3] | Bressan, A.; Constantin, A., Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5, 1-27, (2007) · Zbl 1139.35378 |
| [4] | Camassa, R.; Holm, D. D., An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71, 1661-1664, (1993) · Zbl 0972.35521 |
| [5] | Chen, R.; Lenells, J.; Liu, Y., Stability of the \(\mu\)-Camassa-Holm peakons, J. Nonlinear Sci., 23, 97-112, (2013) · Zbl 1258.35173 |
| [6] | Coclite, G. M.; Holden, H.; Karlsen, K. H., Global weak solutions to a generalized hyperelastic-rod wave equation, SIAM J. Math. Anal., 37, 1044-1069, (2006) · Zbl 1100.35106 |
| [7] | Coclite, G. M.; Holden, H.; Karlsen, K. H., Well-posedness of higher-order Camassa-Holm equations, J. Differential Equations, 246, 929-963, (2009) · Zbl 1186.35003 |
| [8] | Coclite, G. M.; Karlsen, K. H., A note on the Camassa-Holm equation, J. Differential Equations, 259, 2158-2166, (2015) · Zbl 1316.35082 |
| [9] | Constantin, A., On the Cauchy problem for the periodic Camassa-Holm equation, J. Differential Equations, 141, 218-235, (1997) · Zbl 0889.35022 |
| [10] | Constantin, A., On the inverse spectral problem for the Camassa-Holm equation, J. Funct. Anal., 155, 352-363, (1998) · Zbl 0907.35009 |
| [11] | Constantin, A., On the blow-up of solutions of a periodic shallow water equation, J. Nonlinear Sci., 10, 391-399, (2000) · Zbl 0960.35083 |
| [12] | Constantin, A., Global solutions of the Hunter-Saxton equation, SIAM J. Math. Anal., 37, 996-1026, (2005) · Zbl 1108.35024 |
| [13] | Constantin, A., The trajectories of particles in Stokes waves, Invent. Math., 166, 523-535, (2006) · Zbl 1108.76013 |
| [14] | Constantin, A., Particle trajectories in extreme Stokes waves, IMA J. Appl. Math., 77, 293-307, (2012) · Zbl 1250.35150 |
| [15] | Constantin, A.; Escher, J., Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181, 229-243, (1998) · Zbl 0923.76025 |
| [16] | Constantin, A.; Kolev, B., On the geometric approach to the motion of inertial mechanical systems, J. Phys. A, 35, 51-79, (2002) · Zbl 1039.37068 |
| [17] | Constantin, A.; Kolev, B., Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv., 78, 787-804, (2003) · Zbl 1037.37032 |
| [18] | Dai, H. H.; Pavlov, M., Transformations for the Camassa-Holm equation, its high-frequency limit and the sinh-Gordon equation, J. Phys. Soc. Japan, 67, 3655-3657, (1998) · Zbl 0946.35082 |
| [19] | Danchin, R., A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14, 953-988, (2001) · Zbl 1161.35329 |
| [20] | Diperna, R. J.; Lions, P. L., Ordinary differential equations, transport theory, and Sobolev space, Invent. Math., 98, 511-547, (1989) · Zbl 0696.34049 |
| [21] | Escher, J.; Kolev, B., Geodesic completeness for Sobolev \(H^s\)-metrics on the diffeomorphism group of the circle, J. Evol. Equ., 14, 949-968, (2014) · Zbl 1318.58003 |
| [22] | Escher, J.; Kolev, B., Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle, J. Geom. Mech., 6, 335-372, (2014) · Zbl 1308.58005 |
| [23] | Escher, J.; Wunsch, M., Restrictions on the geometry of the periodic vorticity equation, Commun. Contemp. Math., 14, (2012) · Zbl 1248.35157 |
| [24] | Fokas, A.; Fuchssteiner, B., Symplectic structures, their Bäcklund transformations and hereditary symmetries, Physica D, 4, 47-66, (1981) · Zbl 1194.37114 |
| [25] | Fu, Y.; Liu, Z., Non-uniform dependence on initial data for the periodic modified Camassa-Holm equation, NoDEA Nonlinear Differential Equations Appl., 20, 741-755, (2013) · Zbl 1268.35011 |
| [26] | Grafakos, L., Classical Fourier analysis, (2008), Springer-Verlag New York · Zbl 1220.42001 |
| [27] | Gui, G.; Liu, Y., On the global existence and wave-breaking criteria for the two-component Camassa-Holm system, J. Funct. Anal., 258, 4251-4278, (2010) · Zbl 1189.35254 |
| [28] | Himonas, A.; Holmes, J., Hölder continuity of the solution map for the Novikov equation, J. Math. Phys., 54, 1-11, (2013) · Zbl 1298.37051 |
| [29] | Himonas, A.; Kenig, C.; Misiołek, G., Non-uniform dependence for the periodic CH equation, Comm. Partial Differential Equations, 35, 1145-1162, (2010) · Zbl 1193.35189 |
| [30] | Holliman, C., Non-uniform dependence and well-posedness for the periodic Hunter-Saxton equation, Differential Integral Equations, 23, 1159-1194, (2010) · Zbl 1240.35454 |
| [31] | Hunter, J. K.; Saxton, R., Dynamics of director fields, SIAM J. Appl. Math., 51, 1498-1521, (1991) · Zbl 0761.35063 |
| [32] | Hunter, J. K.; Zheng, Y., On a completely integrable nonlinear hyperbolic variational equation, Physica D, 79, 361-386, (1994) · Zbl 0900.35387 |
| [33] | Kato, T., Quasi-linear equations of evolution, with applications to partial differential equations, (Spectral Theory and Differential Equations, Lecture Notes in Mathematics, vol. 448, (1975), Springer Berlin), 25-70 · Zbl 0315.35077 |
| [34] | Kato, T.; Ponce, G., Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41, 203-208, (1988) |
| [35] | Khesin, B., Poisson brackets in hydrodynamics, Discrete Contin. Dyn. Syst., 19, 555-574, (2007) · Zbl 1139.53040 |
| [36] | Khesin, B.; Lenells, J.; Misiołek, G., Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms, Math. Ann., 342, 617-656, (2008) · Zbl 1156.35082 |
| [37] | Lenells, J.; Misiołek, G.; Tig̈lay, F., Integrable evolution equations on spaces of tensor densities and their peakon solutions, Comm. Math. Phys., 299, 129-161, (2010) · Zbl 1214.35059 |
| [38] | Li, Y.; Olver, P., Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations, 162, 27-63, (2000) · Zbl 0958.35119 |
| [39] | Liu, J.; Yin, Z., On the Cauchy problem of a weakly dissipative \(\mu\)-Hunter-Saxton equation, Ann. Inst. Henri Poincaré-AN., 31, 267-279, (2014) · Zbl 1302.35320 |
| [40] | McLachlan, R.; Zhang, X., Well-posedness of a modified Camassa-Holm equations, J. Differential Equations, 246, 3241-3259, (2009) · Zbl 1203.35079 |
| [41] | McLachlan, R.; Zhang, X., Asymptotic blowup profiles for modified Camassa-Holm equations, SIAM J. Appl. Dyn. Syst., 10, 452-468, (2011) · Zbl 1234.35208 |
| [42] | Mu, C.; Zhou, S.; Zeng, R., Well-posedness and blow-up phenomena for a higher order shallow water equation, J. Differential Equations, 251, 3488-3499, (2011) · Zbl 1252.35092 |
| [43] | Olver, P.; Rosenau, P., Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E, 53, 1900-1906, (1996) |
| [44] | Qiao, Z., New hierarchies of isospectral and non-isospectral integrable NLEEs derived from the Harry-Dym spectral problem, Physica A, 252, 377-387, (1998) |
| [45] | Qiao, Z., The Camassa-Holm hierarchy, N-dimensional integrable systems, and algebro-geometric solution on a symplectic submanifold, Comm. Math. Phys., 239, 309-341, (2003) · Zbl 1020.37046 |
| [46] | Qu, C.; Fu, Y.; Liu, Y., Blow-up solutions and peakons to a generalized \(\mu\)-Camassa-Holm integrable equation, Comm. Math. Phys., 331, 375-416, (2014) · Zbl 1304.35141 |
| [47] | Simon, J., Compact sets in the space \(L^p(0, T; B)\), Ann. Mat. Pura Appl., 146, 65-96, (1987) · Zbl 0629.46031 |
| [48] | Tang, H.; Liu, Z., Well-posedness of the modified Camassa-Holm equation in Besov spaces, Z. Angew. Math. Phys., 66, 1559-1580, (2015) · Zbl 1327.35345 |
| [49] | Taylor, M. E., Pseudodifferential operators and nonlinear PDE, (1991), Birkhäuser Boston · Zbl 0746.35062 |
| [50] | Taylor, M. E., Commutator estimates, Proc. Amer. Math. Soc., 131, 1501-1507, (2003) · Zbl 1022.35096 |
| [51] | Xin, Z.; Zhang, P., On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53, 1411-1433, (2000) · Zbl 1048.35092 |
| [52] | Yin, Z., On the structure of solutions to the periodic Hunter-Saxton equation, SIAM J. Math. Anal., 36, 272-283, (2004) · Zbl 1151.35321 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
